On some new inclusion theorems for the eigenvalues of partitioned matrices

Summary Some new results of Gershgorin type for partitioned matrices have been obtained using the so-called departure from normality of the diagonal blocks. This has been shown to improve the existing results at least in the case where diagonal blocks are simultaneously nearly defective and nearly normal. Also a set of Gershgorin-like circles is found such that each of them contains at least one eigenvalue (even if no separation takes place). As a corollary it is shown that every classical Gershgorin circle of a normal matrix contains at least one eigenvalue.